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Keywords:
prime submodules; multiplication modules; distributive lattices; spectral spaces
prime submodules; multiplication modules; distributive lattices; spectral spaces
Summary:
We shall prove that if $M$ is a finitely generated multiplication module and $\mathop {\mathrm Ann}(M)$ is a finitely generated ideal of $R$, then there exists a distributive lattice $\bar{M}$ such that $\mathop {\mathrm Spec}(M)$ with Zariski topology is homeomorphic to $\mathop {\mathrm Spec}(\bar{M})$ to Stone topology. Finally we shall give a characterization of finitely generated multiplication $R$-modules $M$ such that $\mathop {\mathrm Ann}(M)$ is a finitely generated ideal of $R$.
References:
[1] R. Balbes and P. Dwinger: Distributive Lattices. Univ. of Missouri Press, Missouri, 1974. MR 0373985
[2] T. Duraivel: Topology on spectrum of modules. J. Ramanujan Math. Soc. 9 (1994), 25–34. MR 1279099 | Zbl 0835.13001
[3] M. Hochster: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 (1969), 43–60. DOI 10.1090/S0002-9947-1969-0251026-X | MR 0251026 | Zbl 0184.29401
[4] C. P. Lu: $M$-radicals of submodules in modules. Math. Japon. 34 (1989), 211–219. MR 0994584 | Zbl 0706.13002
[5] C. P. Lu: The Zariski topology on the prime spectrum of a module. Houston J. Math. 25 (1999), 417–432. MR 1730888 | Zbl 0979.13005
[6] R. L. McCasland, M. E. Moore and P. F. Smith: Generators for the semimodule of varieties of a free module. Rocky Mountain J. Math. 29 (1999), 1467–1482. DOI 10.1216/rmjm/1181070416 | MR 1743380
[7] H. Simmons: Reticulated rings. J. Algebra 66 (1980), 169–192. DOI 10.1016/0021-8693(80)90118-0 | MR 0591251 | Zbl 0462.13002
[8] P. F. Smith: Some remarks on multiplication modules. Arch. Math. 50 (1998), 223–235. MR 0933916
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